Major airlines have their own medical staff and they often include AMEs.
As these doctors are employed by the airline it makes economic sense for the 'company doc' to do the medical renewal. However this is always optional and individual pilots can elect to use any AME; the costs of a medical are normally reimbursed.
The reason some pilots do not wish to use the 'company doc' is that they feel it could disadvantage them if the company discovered they had an adverse medical condition (unrelated to flying) that could go against them; selection for promotion for example. Others prefer to build up a good relationship with their 'personal' AME that they use for the rest of their professional career.
Some airlines will conduct medicals as part of the recruiting process just to ensure that a pilot meets their standards. However I am sure that the holder of an in-date Class 1 has little to worry about.
HWB